A normal space is a space [typically a topological space] which satisfies one of the stronger separation axioms.
Definitions
A topological space XX is normal if it satisfies:
By Urysohn's lemma this is equivalent to the condition
Often one adds the requirement
- T 1T_1: every point in XX is closed.
[Unlike with regular spaces, T 0T_0 is not sufficient here.]
One may also see terminology where a normal space is any space that satisfies T 4T_4, while a T 4T_4-space must satisfy both T 4T_4 and T 1T_1. This has the benefit that a T 4T_4-space is always also a T 3T_3-space while still having a term available for the weaker notion. On the other hand, the reverse might make more sense, since you would expect any space that satisfies T 4T_4 to be a T 4T_4-space; this convention is also seen.
If instead of T 1T_1, one requires
- R 0R_0: if xx is in the closure of {y}\{y\}, then yy is in the closure of {x}\{x\},
then the result may be called an R 3R_3-space.
Any space that satisfies both T 4T_4 and T 1T_1 must be Hausdorff, and every Hausdorff space satisfies T 1T_1, so one may call such a space a normal Hausdorff space; this terminology should be clear to any reader.
Any space that satisfies both T 4T_4 and R 0R_0 must be regular [in the weaker sense of that term], and every regular space satisfies R 0R_0, so one may call such a space a normal regular space; however, those who interpret ‘normal’ to include T 1T_1 usually also interpret ‘regular’ to include T 1T_1, so this term can be ambiguous.
Every normal Hausdorff space is an Urysohn space, a fortiori regular and a fortiori Hausdorff.
It can be useful to rephrase T 4T_4 in terms of only open sets instead of also closed ones:
- T 4T_4: if G,H⊂XG,H \subset X are open and G∪H=XG \cup H = X, then there exist open sets U,VU,V such that U∪GU \cup G and V∪HV \cup H are still XX but U∩VU \cap V is empty.
This definition is suitable for generalisation to locales and also for use in constructive mathematics [where it is not equivalent to the usual one].
To spell out the localic case, a normal locale is a frame LL such that
- T 4T_4: if G,H∈LG,H \in L are opens and G∨H=⊤G \vee H = \top, then there exist opens U,VU,V such that U∨GU \vee G and V∨HV \vee H are still ⊤\top but U∧V=⊥U \wedge V = \bot.
Examples
We need to show that given two disjoint closed subsets C 1,C 2⊂XC_1, C_2 \subset X, there exist disjoint open neighbourhoods U C 1⊃C 1U_{C_1} \supset C_1 and U C 2⊃C 2U_{C_2} \supset C_2.
Consider the function
d[S,−]:X→ℝ d[S,-] \colon X \to \mathbb{R}
which computes distances from a subset S⊂XS \subset X, by forming the infimum of the distances to all its points:
d[S,x]≔inf{d[s,x]|s∈S}. d[S,x] \coloneqq inf\left\{ d[s,x] \vert s \in S \right\} \,.
If SS is closed and x∉Sx \notin S, then d[S,x]>0d[S, x] \gt 0. Then the unions of open balls
U C 1≔⋃x 1∈C 1B x 1 ∘[12d[C 2,x 1]] U_{C_1} \coloneqq \underset{x_1 \in C_1}{\bigcup} B^\circ_{x_1}[ \frac1{2}d[C_2,x_1] ]
and
U C 2≔⋃x 2∈C 2B x 2 ∘[12d[C 1,x 2]]. U_{C_2} \coloneqq \underset{x_2 \in C_2}{\bigcup} B^\circ_{x_2}[ \frac1{2}d[C_1,x_2] ] \,.
have the required properties. For if there exist x 1∈C 1,x 2∈C 2x_1 \in C_1, x_2 \in C_2 and y∈B x 1 ∘[12d[C 2,x 1]]∩B x 2 ∘[12d[C 1,x 2]]y \in B^\circ_{x_1}[ \frac1{2}d[C_2,x_1] ] \cap B^\circ_{x_2}[ \frac1{2}d[C_1,x_2] ], then
d[x 1,x 2]≤d[x 1,y]+d[y,x 2]